Nonlinear systems and passivity: feedback control, model reduction, and time discretization
Tobias Breiten, Attila Karsai
TL;DR
The paper extends energy‑based control for nonlinear systems by constructing a passive controller that combines an optimal feedback law from the Hamilton‑Jacobi‑Bellman framework with output feedback, yielding energy‑preserving interconnections independent of the plant. It establishes conditions for port‑Hamiltonian realizations, interprets optimality via inverse control, and develops a nonlinear balancing approach that preserves passivity in model order reduction. Numerically, it validates the approach using policy iteration to approximate the value function and a discrete‑gradient discretization to verify passivity in pendulum and Van der Pol examples. This work enables modular, energy‑consistent control and efficient reduced‑order modeling for large‑scale nonlinear passive systems, with practical impact on robust energy‑aware control design.
Abstract
Dynamical systems can be used to model a broad class of physical processes, and conservation laws give rise to system properties like passivity or port-Hamiltonian structure. An important problem in practical applications is to steer dynamical systems to prescribed target states, and feedback controllers combining a regulator and an observer are a powerful tool to do so. However, controllers designed using classical methods do not necessarily obey energy principles, which makes it difficult to model the controller-plant interaction in a structured manner. In this paper, we show that the combination of an optimal feedback law characterized by the Hamilton-Jacobi-Bellman equation and output feedback gives rise to passivity properties of the controller that are independent of the plant structure. Furthermore, we state conditions for the controller to have a port-Hamiltonian realization and show that a model order reduction scheme can be deduced using the framework of nonlinear balanced truncation. To illustrate our results, we numerically realize the controller using the policy iteration and computationally verify passivity via a custom passivity-preserving discrete gradient scheme suitable for a wide class of passive systems.